3.728 \(\int (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=93 \[ \frac {2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {2 a (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]

[Out]

2/7*a*(d*x)^(7/2)*((b*x^2+a)^2)^(1/2)/d/(b*x^2+a)+2/11*b*(d*x)^(11/2)*((b*x^2+a)^2)^(1/2)/d^3/(b*x^2+a)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1112, 14} \[ \frac {2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {2 a (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(2*a*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d*(a + b*x^2)) + (2*b*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(11*d^3*(a + b*x^2))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^{5/2} \left (a b+b^2 x^2\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a b (d x)^{5/2}+\frac {b^2 (d x)^{9/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {2 a (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac {2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 44, normalized size = 0.47 \[ \frac {2 x (d x)^{5/2} \sqrt {\left (a+b x^2\right )^2} \left (11 a+7 b x^2\right )}{77 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(11*a + 7*b*x^2))/(77*(a + b*x^2))

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 26, normalized size = 0.28 \[ \frac {2}{77} \, {\left (7 \, b d^{2} x^{5} + 11 \, a d^{2} x^{3}\right )} \sqrt {d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*((b*x^2+a)^2)^(1/2),x, algorithm="fricas")

[Out]

2/77*(7*b*d^2*x^5 + 11*a*d^2*x^3)*sqrt(d*x)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 45, normalized size = 0.48 \[ \frac {2}{11} \, \sqrt {d x} b d^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{7} \, \sqrt {d x} a d^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*((b*x^2+a)^2)^(1/2),x, algorithm="giac")

[Out]

2/11*sqrt(d*x)*b*d^2*x^5*sgn(b*x^2 + a) + 2/7*sqrt(d*x)*a*d^2*x^3*sgn(b*x^2 + a)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 39, normalized size = 0.42 \[ \frac {2 \left (7 b \,x^{2}+11 a \right ) \left (d x \right )^{\frac {5}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{77 \left (b \,x^{2}+a \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*((b*x^2+a)^2)^(1/2),x)

[Out]

2/77*x*(7*b*x^2+11*a)*(d*x)^(5/2)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 25, normalized size = 0.27 \[ \frac {2 \, {\left (7 \, \left (d x\right )^{\frac {11}{2}} b + 11 \, \left (d x\right )^{\frac {7}{2}} a d^{2}\right )}}{77 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(5/2)*((b*x^2+a)^2)^(1/2),x, algorithm="maxima")

[Out]

2/77*(7*(d*x)^(11/2)*b + 11*(d*x)^(7/2)*a*d^2)/d^3

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^{5/2}\,\sqrt {{\left (b\,x^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(5/2)*((a + b*x^2)^2)^(1/2),x)

[Out]

int((d*x)^(5/2)*((a + b*x^2)^2)^(1/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(5/2)*((b*x**2+a)**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________